When accreting X-ray pulsars (XRPs) undergo bright X-ray outbursts, their luminosity-dependent spectral and timing features can be analyzed in detail. The XRP GRO J1750-27 recently underwent one of such episodes, during which it was observed with NuSTAR and monitored with NICER. Such a data set is rarely available, as it samples the outburst over more than a month at a luminosity that is always exceeding ~5x1037 erg/s. This value is larger than the typical critical luminosity value, where a radiative shock is formed above the neutron star's surface. Our data analysis of the joint spectra confirms the discovery of a deep cyclotron line at a centroid energy of about 44 keV, corresponding to a magnetic field strength of 4.7x1012 G. This value is independently supported by the best-fit physical model for spectral formation in accreting XRPs, which also favors a distance of 14 kpc, in agreement with recent findings. Contrary to theoretical expectations and observational evidence from other similar sources, the pulse profiles as observed by NICER through the outburst raise, peak and decay remain remarkably steady. The NICER spectrum, including the iron K\($\alpha$\) line best-fit parameters, also remain almost unchanged at all probed outburst stages, similar to the pulsed fraction behavior. We argue that all these phenomena are linked and interpret them as resulting from a saturation effect of the accretion column's emission, which occurs in the high-luminosity regime.